NOTE ON THE FUNCTION *x = a?(c-x) + {c (c-x) -6 2 }. 
[From the Quarterly Journal of Pare and Applied Mathematics, vol. xv. (1878), 
pp. 338—340.] 
Starting from the general form 
we have 
^x = 
OLX + ß 
yx + 8 ’ 
<v„ (\ W+I - 1) (ax + /3) + (\ n - A) (- 8x + ß) 
(\ n+1 — l) (yx + 8) + (\ n — X) (yx — a) ’ 
where 
1 _ a 2 + S 2 + 2/37 
+ X a8 — ¡3y 
For the function in question 
cv r _ <P{c-x) 
c (c — x) — b 2 ’ 
(a form which presents itself in the problem of the distribution of electricity upon 
two spheres), the values of a, /3, 7, 8 are 
a = — a 2 , /3 = a 2 c, 7 = — c, 8 = c 2 — b 2 ; 
the equation for A, therefore is 
^ _ 1 a 4 + (c 2 - - 2d 2 c- 
X + \~ a 2 b 2 ’ 
(A 4-1) 2 _ (a 2 + hr — c 2 ) 2 
A a 2 b 2 
or, what is the same thing,